CHAP. XVIII] NUMBER OF PBIMBS. 431 E. Catalan224 obtained the preceding results for the case g=2; then ti is the number of odd quotients [2n/0], fe the number of odd quotients [2n/()9y)], . . . , where ft 7, . . . are the primes >2 and ^rc. L. Gegenbauer225 gave eight formulas, (29)-(36), of the type of Legendre's, a special case of one being where a; ranges over the integers divisible by no prime > \/n, while n(x) is Merten's function (Ch. XIX) and Lk(ri) is the sum of the kih powers of all primes >\/n but ^n. The case & = 0 is Legendre's formula. The case t*l is Sylvester's208 E. Meissel226 computed the number of primes <109. Gegenbauer2260 gave complicated expressions for 0(n), one a generalization of Bougaief s.217 A. Lugli227 wrote 0(n, f) for the number of integers ^n which are divisible by no one of the first i primes px = 2, p2=3, . . . . If i is the number of primes ^ \/n and if s is the least integer such that the number ^(n) of primes ^n, excluding 1, is proved to satisfy This method of computing \[/(n) is claimed to be simpler than that by Legendre or Meissel. J. J. van Laar227a found the number of prunes < 30030 by use of the primes < 1760. C. Hossfeld228 gave a direct proof of the case of the upper signs being due to Meissel.216 F. Rogel229 gave a modification and extension of Meissel' s215 formula. H. Scheffler230 discussed the number of primes between p and q. J. J. Sylvester231 stated that the number of primes >n and <2n is -~- a ao abc if a, 6, ... are the primes ^ \/2n and Hx denotes x when its fractional part . Soc. Sc. Ltege, M&n. No. 1. 225Sitzungsber. Ak. Wiss. Wien (Math.), 89, II, 1884, 841-850; 95, II, 1887, 291-6. ^Math. Annalen, 25, 1885, 251-7. 218«Sitzungsber. Ak. Wiss. Wien (Math,), 94, II, 1886, 903-10. M7Giornale di Mat., 26, 1888, 86-95. 227aNieuw Archief voor Wisk., 16, 1889, 209-214. ^Zeitschrift Math. Phys., 35, 1890, 382-4. M8Math. Annalen, 36, 1890, 304-315. «°Beitrage zur Zahlentheorie, 1891, 187. 231Lucas, The'orie des nombres, 1891, 411-2. Proof by H. W. Curjel, Math. Quest. Educ. Times, 57, 1892, 113.